X6Delta_Hedging

# X6Delta_Hedging - Review Slide 13-1 The Black-Scholes model...

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Slide 13-1 Review

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Slide 13-2 The Black-Scholes model We introduced the Black-Scholes model: We discussed analogies between the Black-Scholes model and the binomial model. We defined the „Greeks“. C S,K, ,r,T, Se N d Ke N d - T -rT ( ) = ( ) ( ) !" " 12 # Like the binomial option pricing formula, the Black-Scholes formula is based on the „replication approach“: \$ = e - " T N(d 1 ) is the position in the underlying asset required to replicate a call. Gamma ( % ): change in \$ when option price increases by \$1 V ega: change in option price when v olatility increases by 1% T heta ( & ): change in option price when t ime to maturity decreases by 1 day R ho ( ): change in option price when interest r ate increases by 1%
Slide 13-3 Understanding the Greeks

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Slide 13-4 Market-Making and Delta-Hedging Chapter 13
Slide 13-5 Outline How to delta hedge a position Error due to gamma Error due to theta Delta hedging and the Black-Scholes model How to delta-gamma hedge a position

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Slide 13-6 What do market makers do? Provide immediacy by standing ready to sell to buyers (at the ask price) and to buy from sellers (at the bid price). Their positions are determined by the order flow from customers. Market makers profit by charging a bid-ask spread. In contrast, proprietary trading relies on an investment strategy that makes a profit based on changes in asset prices. Market makers attempt to hedge their order books in order to avoid potential bankruptcy caused by an adverse price move.
Slide 13-7 Example Suppose a market maker sold a call option. If the market maker would not hedge the option position, he/she would be exposed to price changes of the underlying asset.

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Slide 13-8 The market-maker’s exposure Note : If the stock price does not change, the market maker would earn a profit, due to the time decay of the option.
Slide 13-9 Measuring the market-maker’s exposure The \$ of the option is 0.5824. Therefore, a \$0.75 increase in the stock price would be expected to increase the option value by \$0.75 × 0.5824 = \$0.4368 The actual increase in the option’s value, however, is higher: \$0.4548 This is because \$ increases as the stock price increases. Using the initial \$ at the initial stock price understates the actual price change. Using the resulting \$ at the higher stock price would overstate the actual price change.

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## This note was uploaded on 10/25/2009 for the course 15 15.402 taught by Professor Bergman during the Fall '09 term at MIT.

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X6Delta_Hedging - Review Slide 13-1 The Black-Scholes model...

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